LETTER Three-loop order approach to flat polymerized membranes

A full three-loop calculation by my PhD student Simon Metayer published as a Letter in PRE.
S. Metayer D. Mouhanna and S. Teber, Phys. Rev. E 105 (2022) no.1, L012603 [ arXiv:2109.03796 [cond-mat.stat-mech].]

EVENT Workshop on Multi-Loop Calculations (methods and applications)

Organizers: G. Faye (IAP), S. Teber (LPTHE), P. Tourkine (CERN & LPTHE) and P. Vanhove (IPhT, CEA-Saclay)
Venue: amphi Charpak (Sorbonne University)
Dates: 14 and 15 may 2019
Website: multi-loop-2019.sciencesconf.org

HABILITATION Field theoretic study of electron-electron interaction effects in Dirac liquids

S. Teber, arXiv:1810.08428 [cond-mat.mes-hall]

REVIEW Multi-loop techniques for massless Feynman diagram calculations

A. V. Kotikov and S. Teber, Phys. Part. Nucl. 50 (2019) no.1, 1 [arXiv:1805.05109 [hep-th]]

EVENT Workshop on Multi-Loop Calculations (methods and applications)

Organizers: B. Basso (LPTENS), M. Kompaniets (St Petersbourg University), A. Kotikov (JINR Dubna) and S. Teber (LPTHE)
Venue: amphi Charpak (UPMC)
Dates: 7 and 8 june 2017
Website: multi-loop-2017.sciencesconf.org
Summary and Photos: multi-loop-2017



My research activities focus on the study of interaction effects in low-dimensional quantum systems.

Presently, I am interested in the so-called Dirac materials with a special focus on planar systems such as graphene and graphene-like materials. Graphene is a one-atom thick layer of graphite characterized by gapless bands, strong electron-electron interactions and emergent Lorentz invariance deep in the infra-red. This system is subject to a number of challenging issues at the boundary between condensed matter and high energy physics raised by recent experiments. For example, the influence of interactions on transport and spectral properties of the system and their potential ability to dynamically generate a mass (or gap). The orginality of my approach consists in studying interaction effects in this system starting from the infra-red Lorentz-invariant fixed point where low-energy properties are captured by \((2 + 1)\)-dimensional effective (relativistic) field theories such as the so-called reduced QED and QED\(_3\). These models constitute a nice playground where interaction effects may be studied beyond the common leading order calculations. Such an ambitious task may be achieved via the application of powerful multi-loop techniques originally developed in particle physics and statistical mechanics. Interestingly, the odd dimensionality of space-time together with the (related) presence of Feynman diagrams with non-integer indices brings a lot of novelties (as well as some additional complications) with respect to what is usually known from the study of \((3 + 1)\)-dimensional theories. The study of the fixed point also offers a robust base from which the physics away from the fixed point (which is closer to the experimental situation) may be explored. A striking feature of the results obtained so far is that there seems to be a quantitative agreement between the fixed point physics (relativistic limit with fully retarded interactions) and physics far away from it (non-relativistic limit with instantaneous interactions).

This project involves a very nice collaboration with Anatoly Kotikov from Dubna who is a world-leading expert in the computation of (massless and massive) Feynman diagrams.

Selected publications

  • "Electromagnetic current correlations in reduced quantum electrodynamics,"
    Phys. Rev. D 86 (2012) 025005 [arXiv:1204.5664 [hep-ph]].

  • Due to its peculiar honeycomb lattice, the band structure of (intrinsic) graphene consists of linearly dispersing bands (up to energies of the order of \(1\)eV) crossing at 2 Fermi (or Dirac) points. As was probably first realized long ago by Semenoff (in the case of free fermions) [Phys. Rev. Lett. 53 (1984) 2449], an effective low-energy description then emerges in terms of a simple continuous \(U(1)\) QED-like gauge-field theory of massless Dirac fermions. Upon adding interactions, Gonzáles, Guinea and Vozmediano [Nucl. Phys. B424 (1994) 595] proved the existence of an infra-red (IR) Lorentz invariant fixed point due to the running of the Fermi velocity, \(v\). The later flows to the velocity of light deep in the IR, \(v \rightarrow c\), with a corresponding flow of the coupling constant of graphene to the QED coupling constant: \(\alpha_g = e^2/v \rightarrow \alpha =1/137\). Because \(v \approx c/300 \ll 1\), studies of interaction effects in graphene generally focus on the experimentally relevant non-relativistic limit, \(v/c \rightarrow 0\), where the Coulomb interaction is instantaneous and the physics at the fixed point has been largely overlooked.

    Our work [Phys. Rev. D 86 (2012) 025005] focuses on the study of interaction effects starting from the IR fixed point where \(v=c\) and the interaction is fully retarded. Though a priori mainly of academic interest, the general motivation comes from the fact that interaction effects may be studied in a rigorous and systematic way in this ultra-relativistic limit. Moreover, a full understanding of this limit may allow to extend the developed techniques to experimentally accessible scales. It was then realized in [Phys. Rev. D 86 (2012) 025005] that the effective field theory at the fixed point corresponds to the so-called massless reduced [Gorbar, Gusynin and Miransky, Phys. Rev. D 64 (2001) 105028] or pseudo [Marino, Nucl. Phys. B408 (1993) 551] QED. Reduced QED\(_{d_\gamma,d_e}\) is a quantum field theory describing the interaction of an abelian \(U(1)\) gauge field living in a \(d_\gamma\)-dimensional space-time with a fermionic field living in a reduced space-time of \(d_e\) dimensions (\(d_e \leq d_\gamma\)). The interacting system may therefore be thought of as a physical realization of a "brane"-like universe such as those which are often evoked in particle physics for a larger (and unphysical) number of dimensions. In the case where \(d_\gamma=d_e\), reduced QEDs correspond to usual QEDs. The peculiar case of QED\(_{4,3}\) describes graphene at its fixed point.

    In [Phys. Rev. D 86 (2012) 025005], multi-loop techniques, originally developed in particle physics and statistical mechanics, were applied to massless QED\(_{d_\gamma,d_e}\). This led to the exact computation of the polarization operator up to 2 loops. In the specific case of QED\(_{4,3}\), the polarization operator is related to the optical conductivity of graphene at the fixed point (a quantity which was subject to some debate in the non-relativistic limit, see next item). From the two-loop computation, the first order interaction correction coefficient to the optical conductivity of graphene at the fixed point could be derived: \(\mathcal{C}^* = (92-9\pi^2)/(18\pi)\). Surprisingly, the value of this coefficient, \(\mathcal{C}^* = 0.056\), agrees quantitatively well with the one obtained in the non-relativistic limit (away from the fixed point), \(\mathcal{C} = 0.013\). The value of \(\mathcal{C}^*\) was derived independently in [Phys. Rev. D 87 (2013) no.8, 087701] on the basis of the method of uniqueness, a powerful method for multi-loop computations in higher dimensional theories with conformal symmetry. In the continuity of these research papers, the computation of the two-loop fermion self-energy in reduced QED was performed in [ Phys. Rev. D 89 (2014) no.6, 065038].

  • "Interaction corrections to the minimal conductivity of graphene via dimensional regularization,"
    Europhys. Lett. 107 (2014) 57001 [arXiv:1407.7501 [cond-mat.mes-hall]].

  • The optical conductivity of graphene is an important observable that has been studied by different experimental groups around 2007. Surprisingly, despite the fact that graphene is supposed to be in a strongly coupled regime at experimentally accessible scales, experimental results show very weak deviations of this conductivity with respect to the free fermion result. This has generated extensive theoretical works since 2008. It turns out that contradictory results were obtained: some in agreement with weak deviations such as Mishchenko's result [ Europhys. Lett. 83 , 17005 (2008)] while other works finding larger deviations.

    Our work attempts at clarifying this situation. We do obtain weak deviations in accordance with Mishchenko's analysis. Moreover, we find that the origin of the disagreement lies in subtle renormalisation effects which need to be taken into account whatever regularization method is used. Such clarification is important because, based on the simple example of the optical conductivity, it allows to understand how interaction effects can be systematically taken into account in computing other observables.

    Let us note that our results are valid in the limit of instantaneous Coulomb interaction, \(v/c \rightarrow 0\) (in the non-relativistic limit, away from the Lorentz-invariant fixed point). Technically, we could adapt powerful multi-loop techniques (for semi-massive Feynman diagrams) to this non-relativistic limit. Moreover, from the physics point of view, the value of the first interaction correction coefficient in this limit, \(\mathcal{C} = (19-6\pi)/12 \approx 0.013\), agrees quantitatively well with the one obtained in the ultra-relativistic limit, \(v/c \rightarrow 1\) (with fully retarded interactions), \(\mathcal{C}^* = 0.056\) (see item above).

  • "Critical behaviour of \((2 + 1)\)-dimensional QED: \(1/N_f\)-corrections,"
    Phys. Rev. D 94 (2016) no.5, 056009 [arXiv:1605.01911 [hep-th]].
    Phys. Rev. D 94 (2016) no.11, 114011 [arXiv:1609.06912 [hep-th]].
    arXiv:1902.03790 [hep-th].

  • The understanding of dynamical chiral symmetry breaking (D\(\chi\)SB) in QED\(_3\) is a long-standing problem with now three decades of extensive research. The question of the existence and stability of the critical point (separating massless and massive phases) is central to the vast majority of works. However, very few works address the accurate study of next-to-leading order (NLO) corrections. The reason is simply that such a task is of tremendous difficulty. It is however of utmost importance, especially that the value of the critical fermion flavour number (\(N_c\)) at leading order (LO) is not large.

    In order to appreciate this, let us recall the 2 previous important works on the subject. The first one is from Appelquist, Nash and Wijewardhana, "Critical Behavior in \((2+1\))-dimensional QED," [ Phys. Rev. Lett. 60 2575 (1988)] where the authors provide LO estimate for \( N_c = 32/\pi^2 \approx 3.24 \) (in the Landau gauge). The second one is from Nash, "Higher-order corrections in \((2+1)\)-dimensional QED," [ Phys. Rev. Lett. 62 3024 (1989)] where the author attempts to estimate NLO corrections to \(N_c\). Nash worked in a non-linear gauge and performed a resummation leading to the suppression of the gauge dependence of \(N_c\) at LO yielding: \(N_c = (4/3)(32/\pi^2) \approx 4.32\). His full NLO calculation is however only approximate and has been carried out in the Feynman gauge with no possible discussion of the gauge dependence of \(N_c\) at NLO. In the last 27 years there has been no further substantial progress in understanding NLO corrections in QED\(_3\).

    In the continuity of these research papers, our work [ Phys. Rev. D 94 (2016) no.5, 056009] partially fills this gap by providing an exact computation of all NLO corrections in the Landau gauge. Our second work [ Phys. Rev. D 94 (2016) no.11, 114011] extends these results in two very non-trivial ways. First, all (exact) calculations are carried out for an arbitrary non-local gauge. Second, a Nash-like resummation is performed. We could then confirm the absence of gauge dependence at LO and we could also explicitly prove the strong suppression of the gauge dependence of \( N_c \) at NLO. In our third work: [ arXiv:1902.03790 [hep-th]] we proove the complete cancellation of the gauge dependence of the critical fermion flavour number resulting in: \(Nc=2.8469 \) at NLO. This result is in full agreement with one of Gusynin and Pyatkovskiy [ Phys. Rev. D 94 (2016) no.12, 125009] who used a different method. Thirty years after the seminal work of Nash, these results bring a definite and complete solution to NLO computations in QED\(_3\). They provide order by order fully gauge-invariant methods to compute \(Nc \) and give increasing support for the stability of the critical point. They suggest that D\(\chi\)SB takes place for integer values of \(N\) smaller or equal to 2 (\(N \leq 2\)).

  • "Critical behaviour of reduced QED\(_{4,3}\) and dynamical fermion gap generation in graphene,"
    Phys. Rev. D 94 (2016) no.11, 114010 [arXiv:1610.00934 [hep-th]]

  • The study of dynamical gap generation (excitonic instability) in graphene is a very active field of research for more than a decade now. The problem is very challenging because the effect is non-perturbative and therefore beyond the reach of conventional perturbation theory. Actually, to start with, graphene is a strongly coupled system with bare coupling constant \(\alpha \approx 2.2\). A priori, this strong value should favour the instability. It turns out that there is no experimental evidence for a gap. Theoretically, a very important issue is to compute with high precision the critical coupling constant \(\alpha_c\) which is such that for \(\alpha \gt \alpha_c\) a dynamical gap is generated. Alternatively, the instability manifests for \(N \lt N_c\) where the critical fermion flavour number \(N_c\) is such that \(\alpha_c \rightarrow \infty\) and of importance is also to compute \( N_c \) with high precision (for graphene \( N=2 \)). Starting from the early work of Khveshchenko [ Phys. Rev. Lett. 87 246802 (2001)], this has been done in a number of theoretical papers over the years (in the non-relativistic limit, \(v/c \rightarrow 0\)). The general agreement is that \( \alpha_c = O(1) \) but the precise value of \(\alpha_c\) is still subject to some controversy (some groups finding values smaller than \(2.2\) and others finding values larger than \(2.2\)). Often, the approximations involved are criticised.

    Our work revisits the problem from a completely different angle. We consider the ultra-relativistic limit, \(v/c \rightarrow 1\), which corresponds to graphene at its Lorentz-invariant fixed point. The corresponding effective field theory is called reduced QED\(_{4,3}\). The advantage of considering this limit is that \(\alpha_c\) and \(N_c\) may be derived fully analytically with unprecedented precision (up to NLO). A remarkable feature of our work is a very nice mapping between large-\(N\) QED\(_3\) and reduced QED\(_{4,3}\) which originates from the fact that the photon propagators in both models have the same form. This mapping allowed us to study the critical behaviour of reduced QED\(_{4,3}\) on the basis of our recent exact analysis of D\(\chi\)SB in QED\(_3\) (see item above). So, a first important development brought by our work is that we give more weight to QED\(_3\) as a physical model by directly relating it to a model describing one of the most challenging condensed matter system presently under study.

    From the technical point of view: all calculations are exact and carried out for an arbitrary gauge fixing parameter; a (Nash-like) resummation of the wave-function renormalization is performed which strongly suppresses the gauge dependence of the critical coupling constant, \(\alpha_c\), and critical fermion number, \(N_c\); an additional RPA resummation was performed which is crucial beyond leading order to get non-trivial results. From these results, we could obtain high precision (gauge-invariant) estimates of \(\alpha_c\) and \(N_c\) which are compatible with the semi-metallic behaviour observed experimentally (at the fixed point \(\alpha \approx 1/137 \ll \alpha_c\)).

    A third striking feature of our work is that the value obtained for \(\alpha_c\) at the fixed point is of \(O(1)\) and therefore in good quantitative agreement with the results obtained in the non-relativistic limit, e.g., away from the fixed point, including very good agreement with lattice simulations. This suggest that the study of the fixed point is not only of academic interest and that our model may be an efficient effective field theory model in describing some of the features of actual planar condensed matter physics systems.

See also


Articles in peer-reviewed journals

Refereed conference proceedings


Some recent talks

  • "Landau-Khalatnikov-Fradkin transformation and even zeta-values in Euclidean massless correlators", Rencontre de Physique des Particules 2020, Ecole Polytechnique (Palaiseau, january 2020)

  • "Radiative corrections in planar Dirac liquids", seminar in the group of Prof. Maxim Chernodub, FDP (Tours, january 2019)

  • "Field theoretic study of electron-electron interaction effects in Dirac liquids", Low-dimensional materials: theory, modeling, experiment, LDM2018 (Dubna, july 2018)

  • "New results for the critical fermion flavour number of three-dimensional QED", IV Russian-Spanish Congress: Particle, Nuclear, Astroparticle Physics and Cosmology (Dubna, september 2017) [pdf]

  • "Loops, Triangles and the Optical Conductivity of Graphene", seminar in the group of Prof. Dmitri Kazakov, BLTP, JINR (Dubna, august 2016)

  • "The method of uniqueness and the optical conductivity of graphene: new application of a powerful technique for multi-loop calculations", MQFT2015 International Conference (St Petersburg, september 2015)

  • "Interaction corrections to the optical conductivity of graphene", seminar in the group of Prof. Joerg Schmalian, TKM, KIT (Karlsruhe, may 2015)

  • "Review on interaction corrections to the optical conductivity of graphene", XLIX PNPI Winter School (St Petersburg, march 2015)

  • "Interaction corrections to the optical conductivity of graphene", seminar in the group of Prof. Maria Vozmediano, ICMM, CSIC (Madrid, december 2014)

  • "The method of uniqueness: new application of a powerful technique for multi-loop calculations", ILP thematic day "Methods: Solutions and Challenges" (UPMC, novembre 2014) [pdf]

  • "Reduced Quantum Electrodynamics and Graphene", Exposé AERS, LPTHE (Paris, décembre 2012)

Teaching (in french)

Documents pédagogiques

  • Introduction à la Théorie Quantique des Champs (4P214): polycopié de cours (79 pages) [pdf partiel]

  • Mécanique Analytique (pré-rentrée M1): polycopié de cours (12 pages) [pdf],

  • Mécanique Quantique (pré-rentrée M1): polycopié de cours (12 pages) [pdf],

  • Introduction à l'environnement Unix (4P009): transparents de cours (114 pages) [pdf] (voir [old] pour la version d'origine d'Albert Hertzog et Jacques Lefrère et [new] pour une version plus récente de Jacques Lefrère),

  • Langage C (4P009): transparents de cours (306 pages) [pdf] (voir [old] pour la version d'origine d'Albert Hertzog et Jacques Lefrère et [new] pour une version plus récente de Jacques Lefrère).


Main research interest

  • Interaction effects in low-dimensional quantum systems.
  • Field: condensed matter physics (theoretical).
  • Key words: planar systems (graphene), quantum transport, dynamical symmetry breaking, multi-loop calculations, effective field theories, Schwinger-Dyson equations.

Professional experience and Education

2017: Habilitation (HDR)
2007 - present: Associate Professor (Maître de Conférences) at Sorbonne Université (Université Pierre et Marie Curie), Laboratoire de Physique Théorique et Hautes Energies (LPTHE)
2007: Post-doc at Institut NEEL, Grenoble, France
2004 - 2007: Post-doc at the International Center for Theoretical Physics (ICTP), Trieste, Italy
2002 - 2004: Post-doc at the William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, United-States
1999 - 2002: PhD in Theoretical Solid State Physics at Université Paris XI (Orsay), Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS)
1998 - 1999: Masters in Solid State Physics (Doctoral School of Paris).

Pedagogical tasks (main)

  • Quantum Mechanics (LU3PY111) (2021-present)
  • Quantum Field Theory (MU4PY214) (2019-present)
  • Quantum Mechanics (4P001) (2012-2019)
  • Analytical Mechanics (Master 1 pre-semester course) (2013-present)

Administrative tasks

  • co-responsible/manager (with Sandra Ninet - IMPMC) of the 2nd year physics bachelor at Sorbonne University (approx. 280 students) (2021-present)
  • open science correspondent at the "Faculté des Sciences et Ingénierie" (2020-present)
  • member of the "Groupe d'Experts de l'UFR de Physique" (2015-2021)
  • member of the "Commission de Spécialistes" (2009-2014)


Sorbonne Université (campus Pierre et Marie Curie)
Laboratoire de Physique Théorique et Hautes Energies (LPTHE)
4 place Jussieu, Tour 13-14, 4ème étage, BP 126, 75252 Paris Cedex 05

Office (bureau): 507, tower 13-14, 5th floor (tour 13-14, 5ème étage)

Tel: +33 (0) 1 44 27 28 52

Fax: +33 (0) 1 44 27 73 93

Email: teber [at] lpthe [dot] jussieu [dot] fr

Access plan to LPTHE on Jussieu Campus: [map]